summer student lectures on accelerators (2, 3 ...valdata/rivelatori/acceleratori/...fast extraction...

Elias Métral, Summer Student Course, CERN, 6-7-10-11-12/07/06 1/92

SUMMER STUDENT LECTURES ON ACCELERATORS (2, 3 & 4/5)

Introduction and references (4 slides)Transverse beam dynamics (27)Longitudinal beam dynamics (10)Figure of merit for a synchrotron/collider = Brightness/Luminosity (3)Beam control (12)Limiting factors for a synchrotron/collider ⇒ Collective effects (35)

Space charge (10)Wake field and impedance (12)Beam-beam interaction (7)Electron cloud (6)

EliasElias MMéétraltral ((3 3 45 min = 135 min, 92 45 min = 135 min, 92 slidesslides))

+ Discussion Sessions22

33

[email protected]@cern.chTel.: 72560 or 164809Tel.: 72560 or 164809

1 & 5/5 ⇒ Simone


PURPOSE OF THIS COURSE

Try to give you an overview of the basic concepts & vocabulary ⇒ No mathematics, just physics

…But I would also like you to catch a glimpse of what accelerator physicists are “really” doing, showing you pictures of some recent/current studies


LHC beam

Linear accelerator

Circular accelerator (Synchrotron)

Transfer line

InjectionEjection


SPS tunnel

LHC tunnelTT2 transfer line tunnel

PS tunnelLinac2

PS Booster(after the wall) PS

Vacuum chamber ( = 13 cm here)

What happens to the particles inside the vacuum chamber?


REFERENCES[0] 2005 Summer Student Lectures of O. Bruning,

[http://agenda.cern.ch/askArchive.php?base=agenda&categ=a054021&id=a054021/transparencies]

[1] M. Martini, An Introduction to Transverse Beam Dynamics in Accelerators, CERN/PS 96-11 (PA), 1996, [http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-96-011.pdf]

[2] L. Rinolfi, Longitudinal Beam Dynamics (Application to synchrotron), CERN/PS 2000-008 (LP), 2000, [http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-2000-008.pdf]

[3] Theoretical Aspects of the Behaviour of Beams in Accelerators and Storage Rings: International School of Particle Accelerators of the ‘Ettore Majorana’Centre for Scientific Culture, 10–22 November 1976, Erice, Italy, M.H. Blewett(ed.), CERN report 77-13 (1977)[http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=77-13]

[4] CERN Accelerator Schools [http://cas.web.cern.ch/cas/]

[5] K. Schindl, Space Charge, CERN-PS-99-012-DI, 1999[http://doc.cern.ch/archive/electronic/cern/preprints/ps/ps-99-012.pdf]

[6] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, New York: Wiley, 371 p, 1993 [http://www.slac.stanford.edu/~achao/wileybook.html]

[7] Web site on LHC Beam-Beam Studies [http://wwwslap.cern.ch/collective/zwe/lhcbb/]

[8] Web site on Electron Cloud Effects in the LHC [http://ab-abp-rlc.web.cern.ch/ab-abp-rlc-ecloud/]


TRANSVERSE BEAM DYNAMICS (1/27)

Single-particle trajectory

Circular design orbitOne particle ϑ

In the middle of the vacuum chamber

The motion of a charged particle (proton) in a beam transport channel or a circular accelerator is governed by the LORENTZ FORCE

( )BvEeFrrrr

×+=The motion of particle beams under the influence of the Lorentz force is called BEAM OPTICS



The Lorentz force is applied as a

BENDING FORCE (using DIPOLES) to guide the particles along a predefined ideal path, the DESIGN ORBIT, on which – ideally – all particles should move

FOCUSING FORCE (using QUADRUPOLES) to confine the particles in the vicinity of the ideal path, from which most particles will unavoidably deviate

LATTICE = Arrangement of magnets along the design orbit

The ACCELERATOR DESIGN is made considering the beam as a collection of non-interacting single particles



DIPOLE = Bending magnet

BEAM RIGIDITY [ ] [ ]c/GeV3356.3mT 0pB =ρ

Magnetic field Beam momentumCurvature radius of the dipoles

⇒ A particle, with a constant energy, describes a circle in equilibrium between the centripetal magnetic force and the centrifugal force

N-pole

S-pole

Bg

gx

yBend

magnetDesign

orbit

F

vs

B

( )sρ

Constant force in xand 0 force in y

Beam



PS magnetic field for the LHC beam

0

2

4

6

8

10

150 650 1150 1650 2150

TIME IN THE CYCLE [ms]

INTE

NSI

TY [1

012 pp

p]

0

2000

4000

6000

8000

10000

12000

14000

B [G

auss

]

1st Injection(170) Ejection

(2395)

1 Tesla = 104 Gauss

2nd Injection(1370)

protons per pulse



QUADRUPOLE = Focusing magnet

In x (and Defocusing in y) ⇒ F-type. Permuting

the N- and S- poles gives a D-type

Linear force in x&y

( ) ( ) 0=+′′ sxKsx⇒ : Equation of a harmonic oscillator

From this equation, one can already anticipate the elliptical shapeof the particle trajectory in the phase space (x, x’) by integration

( ) ( ) Constant22 =+′ sxKsx

N-poleS-pole

S-poleN-pole

x

y

Beam

Beam



Along the accelerator K is not constant and depends on s (and is periodic) ⇒ The equation of motion is then called HILL’S EQUATION

The solution of the Hill’s equation is a pseudo-harmonic oscillation with varying amplitude and frequency called BETATRON OSCILLATION

An invariant, i.e. a constant of motion, (called COURANT-SNYDER INVARIANT) can be found from the solution of the Hill’s equation

⇒ Equation of an ellipse (motion for one particle) in the phase space plane (x, x’), with area π a2



The shape and orientation of the phase plane ellipse evolve along the machine, but not its area

are called TWISS PARAMETERS( )sα ( )sβ ( )sγ

( )sx

( )sx′

Courtesy M. Martini

)s(γa

β(s)(s)aα−

( )sx′γ(s)β(s)(s)aα−



Stroboscopic representation or POINCARÉ MAPPING

( )sx

( )sx′

( )0sx

( )0sx′

Depends on the TUNE= Qx = Number of betatron oscillations

per machine revolution



MATRIX FORMALISM: The previous (linear) equipments of the accelerator (extending from s0 to s) can be described by a matrix, M (s / s0), called TRANSFER MATRIX, which relates (x, x’) at s0 and (x, x’) at s

( )( ) ( ) ( )

( )⎥⎦⎤

⎢⎣

⎡′

=⎥⎦

⎤⎢⎣

⎡′ 0

00/

sxsx

ssMsxsx

The transfer matrix over one revolution period is then the product of the individual matrices composing the machine

The transfer matrix over one period is called the TWISS MATRIX

Once the Twiss matrix has been derived the Twiss parameters can be obtained at any point along the machine



In practice, particle beams have a finite dispersion of momentaabout the ideal momentum p0. A particle with momentum p p0 will perform betatron oscillations around A DIFFERENT CLOSED ORBITfrom that of the reference particle

( ) ( ) ( )00

0

ppsD

pppsDsx xx

Δ=−=Δ

is called the DISPERSION FUNCTION

⇒ Displacement of

( )sDx


TRANSVERSE BEAM DYNAMICS (11/27)LHC optics for the Interaction Point (IP) 5 (CMS) in collision

β x,y

[m]

Dx,

y[m

]

s [km]

βx βy Dx Dy Momentum offset = 0



BEAM EMITTANCE = Measure of the spread in phase space of the points representing beam particles ⇒ 3 definitions

1) In terms of the phase plane “amplitude” aq enclosing q % of the particles

2) In terms of the 2nd moments of the particle distribution

3) In terms of σx the standard deviation of the particle distribution in real space (= projection onto the x-axis)

( )%

a ""amplitude of ellipse

qx

q

xddx επ=′∫∫

( )

x

xx

x

βσε σ

2

≡

( ) 222stat >′<−>′<><≡ xxxxxε

[mm mrad] or [μm]

Determinant of the covariance matrix



( )0sx

( )0sx′

particle distribution

particle with “amplitude” ε

0

0

qaεq =a ( )%a qxq ε=

( ) ( ) ( )0%

0 sEs xqxx =εβ

( ) ( )

( )0

%0

sAs

x

qxx

=εγ

Beam envelope

Beam divergence

Emittance

The β-function reflects the size of the beam and depends only on the lattice



NORMALIZED BEAM EMITTANCE

( ) ( )xxxrrnormxσσ εγβε =,

Relativistic factors

⇒ The normalized emittance is conserved during acceleration (in the absence of collective effects…)

ADIABATIC DAMPING: As βr r increases proportionally to the particle momentum p, the (physical) emittance decreases as 1 / p

However, many phenomena may affect (increase) the emittance

An important challenge in accelerator technology is to preserve beam emittance and even to reduce it (by COOLING)

MACHINE mechanical (i.e. from the vacuum chamber) ACCEPTANCE or APERTURE = Maximum beam emittance



FAST EXTRACTION (in 1 turn) in the PS with LHC beam

100 200 300 400 500 600

100

50

50

100

150

�mm�1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fast bumpSlow bump

[mm]

PS circumference [m]

PS Horizontal aperture

PS main magnets

Courtesy O. Berrig

Beam to SPS through TT2 line



BETATRON MATCHING = The phase space ellipses at the injection (ejection) point of the circular machine, and the exit (entrance) of the beam transport line, should be homothetic. To do this, the Twissparameters are modified using quadrupoles. If the ellipses are not homothetic, there will be a dilution (i.e. a BLOW-UP) of the emittance

DISPERSION MATCHING = Dx and D’x should be the same at the injection (ejection) point of the circular machine, and the exit(entrance) of the beam transport line. If there are different, there will be also a BLOW-UP, but due to a missteering (because the beam is not injected on the right orbit)

x

x′

x

x′

x

x′

x

x′

Form of matched ellipse Unmatched beam Filamenting beam Fully filamented

beam

Courtesy D. Möhl



CHROMATICITY = Variation of the tune with the momentum

0/ ppQQ x

x ΔΔ=′

The control of the chromaticity (using a SEXTUPOLE magnet) is very important for 2 reasons

Avoid shifting the beam on resonances due to changes induced by chromatic effects (see later)Prevent some transverse coherent (head-tail) instabilities (see also later)

SEXTUPOLE = 1st nonlinear magnet

Particles approach the reader



In the presence of extra (NONLINEAR) FORCES, the Hill’s equation takes the general form

( ) ( ) ( ) ( )syxPsxsKsx xx ,,=+′′

Any perturbation

Perturbation terms in the equation of motion may lead to UNSTABLEmotion, called RESONANCES, when the perturbating field acts in synchronism with the particle oscillations

A multipole of nth order is said to generate resonances of order n. Resonances below the 3rd order (i.e. due to dipole and quadrupolefield errors for instance) are called LINEAR RESONANCES. The NONLINEAR RESONANCES are those of 3rd order and above



General RESONANCE CONDITIONS PQNQM yx =+

where M, N and P are integers, P being non-negative, |M| + |N| is the order of the resonance and P is the order of the perturbation harmonic

Plotting the resonance lines for different values of M, N, and P in the (Qx, Qy) plane yields the so-called RESONANCE or TUNE DIAGRAM

6.1 6.2 6.3 6.4 6.5Qx

6.1

6.2

6.3

6.4

6.5Qy

xQ

yQ

This dot in the tune diagram is called the

WORKING POINT(case of the PS,

here)



RESONANCE WIDTH = Band with some thickness around every resonance line in the resonance diagram, in which the motion maybe unstable, depending on the oscillation amplitude

STOPBAND = Resonance width when the resonance is linear (i.e. below the 3rd order), because the entire beam becomes unstable if the operating point (Qx, Qy) reaches this region of tune values

DYNAMIC APERTURE = Largest oscillation amplitude which is stable in the presence of nonlinearities

TRACKING: In general, the equations of motion in the presence of nonlinear fields are untractable for any but the simplest situations. Tracking consists to simulate (user computer programs such as MAD) particle motion in circular accelerators in the presence of nonlinear fields



KICK MODEL: Any nonlinear magnet is treated in the “point-like”approximation (i.e. the particle position is assumed not to vary as the particle traverses the field), the motion in all other elements of the lattice is assumed to be linear. Thus, at each turn the local magnetic field gives a “kick” to the particle, deflecting it from its unperturbed trajectory

HENON MAPPING = Stroboscopic repre-sentation of phase-space trajectories (normalised⇒ circles instead of ellipses for linear motion) on every machine turn at the fixed azimuthal position of the perturbation

324.0=xQ 320.0=xQ

252.0=xQ 211.0=xQ

Close to 1/3

Close to 1/4 Close to 1/5

normx

normx′



SEPARATRICES define boundaries between stable motion (bounded oscillations) and unstable motion (expanding oscillations)

The 3rd order resonance is a drastic (unstable) one because the particles which go onto this resonance are lost

(STABLE) ISLANDS: For the higher order resonances (e.g. 4th and 5th) stable motions are also possible in (stable) islands. There are 4 stable islands when the tune is closed to a 4th order resonance and 5 when it is closed to a 5th order resonance



Tune variationTune variation

Phase space Phase space portraitportrait

Simulation for the Simulation for the NEW PS MULTINEW PS MULTI--TURN EXTRACTIONTURN EXTRACTION(Under study)(Under study)

Courtesy M. Giovannozzi

normx

normx′

Examples of transverse gymnastics



Final stage after 20000 turns (about 42 ms for the PS)Final stage after 20000 turns (about 42 ms for the PS)

Slow (Slow (few thousand few thousand turnsturns) bump first (closed ) bump first (closed distortion of the periodic distortion of the periodic orbit)orbit)

Fast (Fast (less than one turnless than one turn) ) bump afterwards (closed bump afterwards (closed distortion of the periodic distortion of the periodic orbit)orbit)


About 6 cm in physical spaceAbout 6 cm in physical space

BBfieldfield ≠≠ 00BBfieldfield == 00At the extraction septum locationAt the extraction septum location

The particles here are extracted through the TT2 transfer line

normx

normx′



Measurements Measurements in the PS in 2004in the PS in 2004

mm

Horizontal beam profile (= projection on the x-axis) during the beam splitting process




SimulationSimulation

Tune variationTune variation

Phase space Phase space portraitportrait

This method can also be used for BEAM INJECTIONThis method can also be used for BEAM INJECTION, generating , generating hollow beams, which is good for space charge (see later)hollow beams, which is good for space charge (see later)


normx

normx′



EMITTANCE EXCHANGE by linear coupling (from skew quadrupoles)

02468

1012

300 320 340 360 380 400

Em

itta

nce

[ μm

] eh 2sigma ev 2sigma h h h

εx(2σ) εy(2σ)

02468

1012

300 320 340 360 380 400

Em

itta

nce

[μ

m]

02468

1012

300 320 340 360 380 400Time [ms]

Em

ittan

ce [

μm

]6.056.106.156.206.256.306.35

250 300 350 400 450Time [ms]

Tun

es

Qx QyQu Qv

A1.0=skewI

A5.1−=skewI

A5.0−=skewI

Theory Measurements

Measurements Measurements in the PS in 2002in the PS in 2002

“Closest tune approach”

Near Qx – Qy = Integer


LONGITUDINAL BEAM DYNAMICS (1/10)

The electric field is used to accelerate or decelerate the particles, and is produced by one or more RF (Radio-Frequency) CAVITIES

]m[s

]m[r

( )∫−

=Δ2/

2/

,g

gs dstsEeE

[eV] [V/m]

Courtesy L. Rinolfi

e-



Six 200 MHz in SS6

13 MHz in SS92 40 MHz in SS78 80 MHz in SS13World Radio Geneva: 88.4 MHz

RF gymnastics

10 MHz RF cavity in Straight Section (SS) 11 of the PS ⇒ For the acceleration

Final power amplifier

2 ferrite loaded cylinders that permit the cavity to be tuned

Accelerating gap



TRANSITION ENERGY: The increase of energy has 2 contradictory effects

An increase of the particle’s velocityAn increase of the length of the particle’s trajectory

According to the variations of these 2 parameters, the revolution frequency evolves differently

Below transition energy: The velocity increases faster than the length ⇒ The revolution frequency increasesAbove transition energy: It is the opposite case ⇒ The revolution frequency decreasesAt transition energy: The variation of the velocity is compensated by the variation of the trajectory ⇒ A variation of energy does not modify the frequency



Sinusoidal voltage applied ( )tVV RFRFRF sinˆ φ= revRF ωω h=

RFV̂

RFV

tRFωφ =RFRFφ tRFRF ωφ =1φ

Synchronous particle

1RF1 sinˆ φVeE =Δ⇒

RFφ

EΔHarmonic numberSYNCHROTRON OSCILLATION

(here, below transition)

Courtesy L. Rinolfi

⇒ Harmonic oscillator for the small amplitudes

BUNCHED beam in a stationary BUCKET

TUNE Qz(<< 1)



Synchrotron oscillation during acceleration(below transition)

0s1 ≠= φφ

Synchronous phase

RFV

RFV̂

RFφ

Above transition, the stable phase is sφπ −

Courtesy L. Rinolfi



RFφ

RFφ

RFV

SEPARATRIX= Limit between the stable and unstable

regions ⇒ Determines the RF BUCKET

Courtesy L. Rinolfi

The number of buckets is given by h

Accelerating bucket



TOMOSCOPE (developed by S. Hancock, CERN/AB/RF)

The aim of TOMOGRAPHYis to estimate an unknown distribution (here the 2D longitudinal distribution) using only the information in the bunch profiles (see Beam control)

[MeV]

[ns]

Surface = Longitudinal EMITTANCE

of the bunch = εL [eV.s]

Surface = Longitudinal ACCEPTANCE of the

bucket

Projection

Projection

Longitudinal bunch profile

Longitudinal energy profile



Longitudinal BUNCH SPLITTING

[MeV]

[ns]

Courtesy S. Hancock

Examples of RF gymnastics



Triple bunch splitting

50 ns/div1 tr

ace

/ 356

revo

lutio

ns (~

800

μs)

Courtesy R. Garoby



BUNCH ROTATION (with ESME)

Parabolic distribution

ns50=bτ

%8.30

=Δpp

ns2.9deg1 =


FIGURE OF MERIT FOR A SYNCHROTRON / COLLIDER: BRIGHTNESS / LUMINOSITY (1/3)

(2D) BEAM BRIGHTNESSyx

IBεεπ 2=

Transverse emittances

Beam current

MACHINE LUMINOSITY event

ondeventsNLσ

sec/=Number of events per second

generated in the collisions

Cross-section for the event under study

The Luminosity depends only on the beam parameters ⇒ It is independent of the physical reaction

[cm-2 s-1]

Reliable procedures to compute and measure



⇒ For a Gaussian (round) beam distribution

FfMNLn

rrevb*

2

4 βεπγ=

Number of particles per bunch

Number of bunches per beam

Revolution frequency

Relativistic velocity factor

Normalized transverse beam

emittanceβ-function at the collision point

Geometric reduction factor due to the crossing angle

at the IP

PEAK LUMINOSITY for ATLAS&CMS in the LHC = 1-2-34 scm10=peakL



INTEGRATED LUMINOSITY ( ) dttLLT

∫=0

int

⇒ The real figure of merit = events ofnumber int =eventL σ

LHC integrated Luminosity expected per year: [80-120] fb-1

Reminder: 1 barn = 10-24 cm2

and femto = 10-15


BEAM CONTROL (1/12)

New CERN Control Centre (CCC) at Prevessin since March 2006

Island for the PS complex

Island for the Technical Infrastructure + LHC

cryogenics

Island for the LHC

Island for the SPS

ENTRANCE


BEAM CONTROL (2/12)WALL CURRENT MONITOR = Device used to measure the instantaneous value of the beam current

Output

Beam

Ferrite rings

Gap

A Wall Current Monitor

Longitudinal bunch profiles for a LHC-type beam

in the PSCourtesy

J. Belleman

Induced or wall current

LoadFor the vacuum + EM shielding

⇒ High-frequency

signals do not see the short

circuit


BEAM CONTROL (3/12)

WALL CURRENT MONITORIN SS3 OF THE PS


BEAM CONTROL (4/12)

(Transverse) beam POSITION PICK-UP MONITOR

LR

LR

UUUUwx

+−=

2

⇒ Horizontal beam orbit measurement in the PS

-8

-6

-4

-2

0

2

4

6

8

3 13 23 33 43 53 63 73 83 93

Straight Section

[mm

]

6 spikes observed as

25.6≈xQ

Beam

LU RU

x

s Beam

0 wCourtesy H. Koziol


BEAM CONTROL (5/12)

POSITION PICK-UP IN SS5 OF THE PS

BEAM LOSS MONITOR

Correction QUADRUPOLE

Correction DIPOLE


BEAM CONTROL (6/12)

FAST WIRE SCANNER

- 30 - 20 - 10 10 20 30POSITION@mmD

200

400

600

800

1000

HORIZONTAL PROFILE

Gaussianfit

( )

x

xx

x

βσε σ

2

≡

⇒ Measures the transverse beam profiles by detecting the particles scattered from a thin wire swept rapidly through the beam

Courtesy S. Gilardoni


BEAM CONTROL (7/12)

FAST WIRE SCANNER IN SS54 OF THE PS

SCINTILLATORS⇒ Producing light

PHOTOMULTIPLIER⇒ Converting light into

and electrical signalMotor of the wire scanner

Light conductors


BEAM CONTROL (8/12)

BEAM LOSS MONITOR

3.42 1013 ppp @ 14 GeV/c(Monday 27/09/04 12:47)

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90 100

BLM NUMBER

LOSS

ES (8

bits

: fro

m 0

to 2

55)

Saturation at 255Could be much

higher

Detector = ACEM (Aluminum Cathod Electron Multiplier), placed at the beginning of the SS ⇒ Similar to a Photomultiplier and it is cheap, robust and radiation resistant

PS recordintensity (CNGS

beam)


BEAM CONTROL (9/12)CHROMATICITY ⇒ Time evolution for the linear Head-Tail phase shift

1st trace = turn 1

Last trace = turn 308

Every turn shown

ns8.24 == tb στ

turnsSPS308=sT

0=yξ

Head and Tail remain in phase

after a kick

Signal at a Pick-Up


BEAM CONTROL (10/12)

ns8.24 == tb στ

turnsSPS308=sT

05.0=yξ

1st trace = turn 1

Last trace = turn 308

Every turn shown



turn 2

Head and Tail in phase



turn 150

~ Maximum phase difference between

Head and Tail


LIMITING FACTORS FOR A SYNCHROTRON / COLLIDER ⇒ COLLECTIVE EFFECTS (1/35)

Space charge

2 particles at rest or travelling

Many charged particles travellingin an unbunchedbeam with circular cross-section

Courtesy K. Schindl

βr



Quadrupole(F-type)

FocusingLinear

Uniform

DefocusingLinear

Gaussian

DefocusingNon-Linear

Proton

Beam

Courtesy K. Schindl


Case of a quasi-parabolic (2nd order) function extending up to 3.2 σ


2 0 2

x��

20

2y��

1

0.5

0

0.5

1

Fx � 2�0� �2��e2n0

0

2

σ/x

σ/y

Spac

e ch

arge

fo

rce

[a.u

.]

- 4 - 3 - 2 - 1 1 2 3 4x�s0.1

0.2

0.3

0.4s gHxL

GaussianGaussian22ndnd orderorder1515thth orderorder

n = 2 ⇔ 3.2σn = 15 ⇔ 6σ

Transverse beam profiles

LHC collimators setting

Rms beam size


64.3086 64.3088 64.309 64.3092 64.3094 64.3096 64.3098

59.3186

59.3188

59.319

59.3192

59.3194

59.3196

59.3198

xQ

yQ

Low-intensity working point

2D TUNE FOOTPRINT⇒ INCOHERENT

(single-particle) tunes

3D view of the 2D tune footprint



shift tunecharge-spaceLinear

Δ 20

=

−∝ normrmsrr

bNεγβ

Small-amplitudeparticles

Large-amplitudeparticles


The longitudinal variation (due to synchrotron oscillations) of the transverse space-charge force fills the gap until the low-intensity working point


2D tune footprint 3D tune footprint

64.3086 64.3088 64.309 64.3092 64.3094 64.3096 64.3098

59.3186

59.3188

59.319

59.3192

59.3194

59.3196

59.319864.3086 64.3088 64.309 64.3092 64.3094 64.3096 64.3098

59.3186

59.3188

59.319

59.3192

59.3194

59.3196

59.3198

xQ

yQ

xQ

yQ

xQ

yQ


Large-(synchrotron)

amplitudeparticles


Qx

Qy

hQ

vQ Regime where continuous loss occurs ⇒ Due to longitudinal motion

Qx

QyvQ

hQ

Regime of loss-free core-emittance blow-up

Particles diffuse into a halo

xQ xQ

yQyQ Regime of loss-free core-emittance blow-up

Regime where continuous loss occurs Due to longitudinal motion

254 =xQ

Interaction with a resonanceInteraction with a resonance


2D Gaussian bunch


5.9 6.1 6.2Qx

5.9

6.1

6.2

QyvQ

hQ

Case 3

11.6≈xQ

24.6≈yQ

-40 -20 20 40

200

400

600

800

[mm]

5.9 6.1 6.2Qx

5.9

6.1

6.2

Qy

Case 1

16.6≈xQ

24.6≈yQ

vQ

hQ

Horizontal bunch profile+ Gaussian fit

Regime of loss-free core-emittance blow-up

xQ

xQ

yQ

yQ



180 ms

1200 ms

Undershoot due to the measurement

device

6.27=xQ

[ms] Time

[ ]1010×bN

Octupole ON (40 A)

170 300 1200

50% of losses

Regime where continuous loss occurs Due to longitudinal motion




INTENSITYINTENSITY--DEPENDENT EMITTANCE EXCHANGE (DEPENDENT EMITTANCE EXCHANGE (Montague resonanceMontague resonance) ) from the nonlinear space charge from the nonlinear space charge near near QQxx = = QQyy

5

10

15

20

25

30

35

40

6.15 6.17 6.19 6.21 6.23 6.25

Horizontal tune

3D simulation results (IMPACT code from R.D. Ryne) for the PS in the case where the synchrotron period is much larger than the crossing time

6.21=yQ

Fit by formulae similar to the ones with linear coupling

( )σε 2,normx

( )σε 2,normy


“High-intensity” bunch ⇒ Bucket separatrix with/without space charge below transition

[ns]

[MeV]

Courtesy S. Hancock


Space charge adds to RF above transition

⇒ Increases the bucket


Wake field and impedance

Wake fields = Electromagnetic fields generated by the beam interacting with its surroundings (vacuum pipe, etc.)

Energy lossBeam instabilities Excessive heating

For a collective instability to occur, the beam must not be ultra-relativistic, or its environment must not be a perfectly conducting smooth pipe

Impedance (Sessler&Vaccaro) = Fourier transform of the wake field

As the conductivity, permittivity and permeability of a material depend in general on frequency, it is usually better (or easier) to treat the problem in the frequency domain




100 10000 1. �106 1. �108 1.�1010f�Hz

100

10000

1. �106

1. �108

1.�1010Zy��m� 1 meter long round LHC collimator

Classical thick-wall

mm2=b∞=Cd

mn17 Ω=Cuρ

μm5=Cud

mμ10 Ω=Cρ

COMPARISON ZOTTER2005COMPARISON ZOTTER2005--BUROV&LEBEDEV2002BUROV&LEBEDEV2002

Im

BL’s results (real and imag. parts) in black: dots without and lines with copper coating

Re



BUNCHED-BEAM COHERENT INSTABILITIESCOUPLED-BUNCH MODES

M = 8 bunches ⇒8 modes n (0 to 7)

possible

Reminder: 2 possible modes with 2 bunches (in phase or out of phase)

Amplitude

Amplitude

n = 1

n = 4

Phase shift between adjacent bunches for the different coupled-bunch modes

2 π R

Bunch treated as a Macro-Particle

F. Sacherer

Mn /2 πφ =Δ



COHERENT frequency (or tune) and INSTABILITY RISE-TIME

We are looking for coherent motions proportional to tωje

iR jωωω += ⇒ ( ) τωωωt

tjtjjtωj eeee RiR == +

where τ is the instability rise time [in s]:

iωτ 1−=

Coherent (angular) frequency

rev, ωωR

cohyQ =Coherent tune



HEAD-TAIL (Single-bunch) MODES: Low intensity

Defined by 2 modes (as there are 2 degrees of freedom, amplitudeand phase) ⇒ Azimuthal mode m and radial mode q

The basic mathematical tool used for the mode representation of the beam motion is the VLASOV EQUATION, using a distribution of particles instead of the single-particle formalism

The Vlasov equation (in its most simple form) is nothing else but a collisionless Boltzmann equation, or an expression for the LIOUVILLE’S CONSERVATION OF PHASE SPACE DENSITY*

*According to the Liouville’s theorem, the particles, in a non-dissipative system of forces, move like an incompressible fluid in phase space



t

S00

t

S02

t

S04

t

S11

t

S13

t

S22

t

S24

t

S33

Signal at position Pick-Up predicted by Theory(several turns superimposed)

Head of the bunch Tail of the

bunch

Mode mq ⇒ q nodes


⇒ Standing-wave pattern


SS2424

τ

Head of the bunch

Tail of the bunch



Observations in the PS in 1999 (20 revolutions superimposed)

7== qm5== qm4== qm

8== qm 10== qm6== qm

Time (20 ns/div)


Σ, ΔR, ΔV signals

Time (10 ns/div)~ 700 MHz

Head stable Tail unstable

HEAD-TAIL (Single-bunch) MODES: High intensity ⇒ Transverse Mode-Coupling instability


Observation in the PS in 2000

⇒ Travelling-wave pattern along the bunch


1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

Head Tail

Observation in the SPS in 2003


⇒ Travelling-wave pattern


Similar phenomena in the longitudinal plane ⇒ Observation of an unstable bunch (sextupolar instability) in the PS Booster in 2000

[ns]

[MeV]

Courtesy S. Hancock




Stabilization methodsLandau damping

Feedbacks

Linear coupling between the transverse planes (from skew quadrupoles)

FeedbacksFeedbacks

Kicker

Electronics

Beam0s 1s

Pick-ups

Courtesy O. Bruning Stabilization when the coherent tune is inside the incoherent tune spread (due to octupoles….)


Beam-beam interaction

Incoherent beam-beam effects⇒ Lifetime + dynamic aperture

PACMAN effects ⇒ Bunch to bunch variation

Coherent beam-beam effects⇒ Beam oscillations and instabilities


ATLAS

CMS

LHC-BALICE

LAYOUT OF THE LHCCourtesy W. Herr

High-luminosity ⇒ Higgs boson

Ions ⇒ New phase of matter expected: Quark-Gluon

Plasma (QGP)

Beauty quark physics

+ TOTEM⇒ Measure the total proton-proton cross-section and study

elastic scattering and diffractive physics



CROSSING ANGLE ⇒ To avoid unwanted collisions, a crossingangle is needed to separate the 2 beams in the part of the machine where they share a vacuum chamber

30 long-range interactions around each IP ⇒ 120 in total

Separation: 9 σ

285 μrad

Courtesy W. Herr

Courtesy W. Herr


2D tune footprint for nominal LHC parameters in collision. Particles up to amplitudes of 6 σ are included



shift tunebeam-beamLinear =∝ normrms

bNε

ξ

xQ

yQ

No 1 / r2 term as for

space charge, as the 2 beams are moving in

opposite direction

Due to head-on

Due to long-range

Courtesy W. Herr


PACMAN BUNCHESLHC bunch filling not continuous: Holes for injection, extraction, dump…2808 bunches out of 3564 possible bunches ⇒ 1756 holesHoles will meet holes at the IPsBut not always… a bunch can meet a hole at the beginning and end of a bunch train


Courtesy W. Herr


Bunches which do not have the regular collision pattern have been named PACMAN bunches ⇒ integrated beam-beam effect

Only 1443 bunches are regular bunches with 4 head-on and 120 long range interactions, i.e. about half of the bunches are not regular

The identification of regular bunches is important since measurements such as tune, orbit or chromaticity should be selectively performed on them

SUPERPACMAN bunches are those who will miss head-on interactions• 252 bunches will miss 1 head-on interaction• 3 will miss 2 head-on interactions

ALTERNATE CROSSING SCHEME: Crossing angle in the verticalplane for IP1 and in the horizontal plane for IP5 ⇒ The purpose is to compensate the tune shift for the Pacman bunches



COHERENT BEAM-BEAM EFFECT


A whole bunch sees a (coherent) kick from the other (separated) beam ⇒ Can excite coherent oscillations

All bunches couple together because each bunch "sees" many opposing bunches ⇒ Many coherent modes possible!

Courtesy W. Herr



σ-mode (in phase) is at unperturbed tune

π-mode (out of phase) is shifted by 1.1 – 1.3 ξ

Incoherent spread between [0.0,1.0] ξ

⇒ Landau damping is lost(coherent tune of the π-mode not inside the incoherent tune spread)

Beam-beam modes and tune spread

ξ

π - mode

σ - mode

Courtesy W. Herr

σ - mode

π - mode

For 2 colliding bunches

It can be restored when the symmetry

between the 2 beams is broken


Electron cloud


Schematic of electron-cloud build up in the LHC beam pipe during multiple bunch passages, via photo-emission (due to synchrotron radiation) and secondary emission

Courtesy F. Ruggiero

The LHC is the 1st proton storage ring for which synchrotron radiation becomes a noticeable effect



Simulations of electron-cloud build-up along 2 bunch trains (= 2 batches of 72 bunches) of LHC beam in SPS dipole regions

Courtesy D. Schulte

SATURATION ⇒ Stops at ~ the

neutralization density, where the attractive force from the

beam is on average balanced by the space charge repulsion

of the electron cloud



Nominal beam for LHC seen on a Pick-Up in the TT2 transfer line

yx

Time (200 ns/div)

⇒ Confirmation that the electron cloud build-up is a single-pass effect

Baseline drift in an electrostatic Pick-Up due to the presence of electrons



yx

Time (200 ns/div)

Same as before but with a solenoidal field (~ 50-100 G) due to ~70 windings before and after the 25 cm long Pick-Up device

~70 windings ~70 windings



Schematic of the single-bunch (coherent) instability induced by an electron cloud

Courtesy G. Rumolo

Single-Bunch Instability From ECloud.mpegMOVIE

Courtesy G. Rumoloand F. Zimmermann



Incoherent effects induced by an electron cloud

x

s

Tail of the bunch Head of the bunch

Electron Cloud footprint

Space Charge footprint Courtesy G. Franchetti

Does not disappear at high energy!

This spread must also be accommodated in the tune diagram

without crossing dangerous resonance lines…


APPENDIX A: SUMMARY OF LECTURE 2

Design orbit in the centre of the vacuum chamberLorentz forceDipoles (constant force) ⇒ Guide the particles along the design orbitQuadrupoles (linear force) ⇒ Confine the particles in the vicinity of the design orbitBetatron oscillation in x (and in y) ⇒ Tune Qx (and Qy) >> 1Twiss parameters define the ellipse in phase space (x, x’ = d x / d s)β-function reflects the size of the beam and depends only on the latticeBeam emittance must be smaller than the mechanical acceptanceHigher order multipoles from imperfections (nonlinear force) ⇒ Resonances excited in the tune diagram and the working point (Qx, Qy) should not be close to most of the resonancesNonlinearities reduce the acceptance ⇒ Dynamic apertureInjection and extractionBetatron and dispersion matching (between a circular accelerator and a transfer line)

( )BvEeFrrrr

×+=


RF cavities are used to accelerate (or decelerate) the particlesTransition energy and sinusoidal voltage ⇒Harmonic number = Number of RF buckets (stationary or accelerating)Bunched beam (instead of an unbunched or continuous beam)Synchrotron oscillation around the synchronous particle in z⇒ Tune Qz << 1Stable phase s below transiton and π – s above transitionEllipse in phase space (Δt, ΔE)Beam emittance must be smaller than the bucket acceptanceBunch splittings and rotation very often usedFigure of merit for a synchrotron/collider = Brightness/LuminosityLongitudinal bunch profile from a wall current monitorTransverse beam orbit from beam position pick-up monitorsTransverse beam profile from a fast wire scannerBeam losses around the accelerator from beam loss monitorsChromaticity from linear Head-Tail phase shift

( )BvEeFrrrr

×+=

APPENDIX B: SUMMARY OF LECTURE 3


(Direct) space charge = Interaction between the particles (without the vacuum chamber) ⇒ Coulomb repulsion + magnetic attraction

Tune footprint in the tune diagram ⇒ Interaction with resonancesDisappears at high energyReduces the RF bucket below transition and increases it above

Wake fields = Electromagnetic fields generated by the beam interacting with its surroundings (vacuum pipe, etc.) ⇒ Impedance = Fourier transform of the wake field

Bunched-beam coherent instabilities• Coupled-bunch modes• Single-bunch or Head-Tail modes (low and high intensity)

Beam stabilization• Landau damping• Feedbacks• Linear coupling between the transverse planes

APPENDIX C: SUMMARY OF LECTURE 4 (1/2)


Beam-Beam = Interaction between the 2 counter-rotating beams ⇒ Coulomb repulsion + magnetic repulsion

Crossing angle, head-on and long-range interactionsTune footprint in the tune diagram ⇒ Interaction with resonancesDoes not disappear at high energyPACMAN effects ⇒ Alternate crossing schemeCoherent modes ⇒ Possible loss of Landau damping

Electron cloudElectron cloud build-up ⇒ Multi-bunch single-pass effectCoherent instabilities induced by the electron cloud

• Coupled-bunch• Single-bunch

Tune footprint in the tune diagram ⇒ Interaction with resonancesDoes not disappear at high energy

APPENDIX C: SUMMARY OF LECTURE 4 (2/2)

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